Multiple positive solutions of singular-Laplacian problems by variational methods

p

-LAPLACIAN PROBLEMS BY VARIATIONAL METHODS

Received 20 July 2004

We obtain multiple positive solutions of singularp-Laplacian problems using variational methods. The techniques are applicable to other types of singular problems as well.

1. Introduction

We consider the singular quasilinear elliptic boundary value problem

−∆pu=a(x)u−γ+λ f(x,u) inΩ,

u >0 inΩ, u=0 on∂Ω,

(1.1)

whereΩis a boundedC2_{domain inR}n_,n≥1,∆

pu=div(|∇u|p−2∇u) is thep-Laplacian,

1< p <∞,a≥0 is a nontrivial measurable function,γ >0 is a constant,λ >0 is a param-eter, and f is a Carath´eodory function onΩ×[0,∞) satisfying

sup (x,t)∈Ω×[0,T]

_{f(x,t)<∞ ∀T >0. (1.2)}

The semilinear case p=2 withγ <1 and f =0 has been studied extensively in both bounded and unbounded domains (see [5,6,7,10,11,12,14,20] and their references). In particular, Lair and Shaker [11] showed the existence of a unique (weak) solution when Ωis bounded anda∈L2_{(Ω). Their result was extended to the sublinear case f(t)=t}β_,

0< β≤1 by Shi and Yao [15] and Wiegner [18]. In the superlinear case 1< β <2∗−1 and for smallλ, Coclite and Palmieri [4] obtained a solution whena=1 and Sun et al. [16] obtained two solutions using the Ekeland’s variational principle for more generala’s. Zhang [19] extended their multiplicity result to more general superlinear terms f(t)≥0 using critical point theory on closed convex sets. The ODE casen=1 was studied by Agarwal and O’Regan [1] using fixed point theory and by Agarwal et al. [2] using varia-tional methods. The purpose of the present paper is to treat the general quasilinear case p∈(1,∞),γ∈(0,∞), and f is allowed to change sign. We use a simple cutoffargument and only the basic critical point theory. Our results seem to be new even forp=2.

First we assume

(H1)∃ϕ≥0 inC10(Ω) andq > nsuch thataϕ−γ∈Lq(Ω).

This does not requireγ <1 as usually assumed in the literature. For example, whenΩis the unit ball,a(x)=(1− |x|2₎σ_{,σ≥0, andγ < σ+ 1/n, we can takeϕ(x)=1− |x|}2_and

q <1/(γ−σ) (resp.,qwith no additional restrictions) ifγ > σ(resp.,γ≤σ).

Theorem1.1. If (H1) and (1.2) hold and f ≥0, then∃λ0>0such that problem (1.1) has

a solution∀λ∈(0,λ0).

Corollary1.2. Problem (1.1) with f =0has a solution if (H1) holds.

Next we allowf to change sign, but strengthen (H1) to (H2)a∈L∞(Ω) witha0:=infΩa >0 andγ <1/n.

This implies thataϕ−γ∈Lq_{(Ω) for anyϕwhose interior normal derivative∂ϕ/∂ν>0 on}

∂Ωandq <1/γ.

Theorem1.3. If (H2) and (1.2) hold, then∃λ0>0such that problem (1.1) has a solution

∀λ∈(0,λ0).

Finally we assume that f isC1_{int, satisfies}

_ft(x,t)≤C

tr−2_{+ 1 (1.3)}

for some 2≤r < p∗, andp-superlinear:

0< θF(x,t)≤t f(x,t), tlarge (1.4)

for someθ > p. Here p∗=np/(n−p) (resp.,∞) if p < n(resp., p≥n) is the critical Sobolev exponent andCdenotes a generic positive constant.

Theorem 1.4. If p≥2, (H1), (1.3), and (1.4) hold, and f ≥0, then ∃λ0>0such that

problem (1.1) has two solutions∀λ∈(0,λ0).

Theorem1.5. If p≥2and (H2), (1.3), and (1.4) hold, then∃λ0>0such that problem

(1.1) has two solutions∀λ∈(0,λ0).

2. Preliminaries on thep-Laplacian

Consider the problem

−∆pu=g(x) inΩ,

u=0 on∂Ω. (2.1)

Proposition2.1. Ifg∈Lq_{(Ω)for someq > n, then (2.1) has a unique weak solutionu∈}

C1

0(Ω). If, in addition,g≥0is nontrivial, then

Proof. The existence of a unique solutionu∈W01,p(Ω) is well-known. The problem

−∆v=g(x) inΩ,

v=0 on∂Ω (2.3)

has a unique solutionv∈W2,q_(Ω)C1,α_{(Ω),α=1−n/q. Thenusatisfies}

div|∇u|p−2_{∇u−G(x)=0 inΩ,}

u=0 on∂Ω, (2.4)

whereG= ∇v∈Cα_{(Ω), anduis bounded by Guedda and V´eron [8] sinceq > n/ p if}

p≤n, sou∈C1

0(Ω) by Lieberman [13]. The rest now follows from V´azquez [17].

3. Proofs of Theorems1.1and1.3

Proof ofTheorem 1.1. Sincea∈Lq_{(Ω) by (H}

1), the problem

−∆pv=a(x) inΩ,

v=0 on∂Ω (3.1)

has a unique positive solutionv∈C1

0(Ω) with∂v/∂ν>0 on∂ΩbyProposition 2.1. Then infΩ(v/ϕ)>0 and henceav−γ∈Lq(Ω). Fix 0< ε≤1 so small thatu:=ε1/(p−1)v≤1. Then

−∆pu−a(x)u−γ−λ f(x,u)≤ −(1−ε)a(x)≤0, (3.2)

souis a subsolution of (1.1). Sinceau−γ∈Lq_{(Ω), the problem}

−∆pu=a(x)u(x)−γ+ 1 inΩ,

u=0 on∂Ω (3.3)

has a unique solutionu∈C1

0(Ω) byProposition 2.1, andu≥usince

−∆pu≥a(x)≥εa(x)= −∆pu. (3.4)

Then

−∆pu−a(x)u−γ−λ f(x,u)≥1−λ sup x∈Ω,t≤maxΩu

f(x,t), (3.5)

Let

gλ,u(x,t)=

        

a(x)u(x)−γ+λ fx,u(x), t > u(x) a(x)t−γ+λ f(x,t), u(x)≤t≤u(x) a(x)u(x)−γ+λ fx,u(x), t < u(x),

Gλ,u(x,t)=

t

0gλ,u(x,s)ds,

Φλ,u(u)=

Ω|∇u|

p_−pG

λ,u(x,u), u∈W01,p(Ω).

(3.6)

Since

0≤gλ,u(x,t)≤a(x)u(x)−γ+λ sup x∈Ω,t≤maxΩu

f(x,t), ∀(x,t)∈Ω×R, (3.7)

andau−γ∈Lq_{(Ω), Φ}

λ,uis bounded from below and has a global minimizeru0, which

then is a solution of (1.1) in the order interval [u,u].

Proof ofTheorem 1.3. The problem

−∆pv=a0 inΩ,

v=0 on∂Ω (3.8)

has a unique positive solutionv∈C1

0(Ω) with∂v/∂ν>0 on∂Ω. Fix 0< ε <1 so small that

u:=ε1/(p−1)_{v≤1. Then}

−∆pu−a(x)u−γ−λ f(x,u)≤ −(1−ε)a0+λ sup x∈Ω,t≤maxΩu

_{f(x,t), (3.9)}

so∃λ0>0 such thatuis a subsolution of (1.1)∀λ∈(0,λ0). The rest of the proof now

proceeds as above.

4. Proofs of Theorems1.4and1.5

Proof ofTheorem 1.4. Define a Carath´eodory function onΩ×Rby

gλ(x,t)=

  

a(x)t−γ+λ f(x,t), t≥u(x)

a(x)u(x)−γ+λ fx,u(x), t < u(x) (4.1)

and consider the problem

−∆pu=gλ(x,u) inΩ,

u=0 on∂Ω. (4.2)

Every solution of (4.2) is≥uand hence also a solution of (1.1). By (1.3),

0≤gλ(x,t)≤a(x)u(x)−γ+λC t+

r−1

so solutions of (4.2) are the critical points of theC1_functional

Φλ(u)=

Ω|∇u|

p_−pG

λ(x,u), u∈W 1,p

0 (Ω), (4.4)

whereGλ(x,t)=

t

0gλ(x,s)ds. Sinceu0solves

−∆pu=gλ,ux,u0(x) inΩ,

u=0 on∂Ω (4.5)

andgλ,u(·,u0(·))∈Lq(Ω) by (3.7),u0∈C10(Ω) byProposition 2.1. Note that, with a pos-sibly smallerλ0, 2uis also a supersolution of (1.1)∀λ∈(0,λ0). We assume thatu0is the global minimizer of the corresponding functionalΦλ,2ualso, for otherwise we are done.

Since

u0≤u <2u inΩ, ∂u0/∂ν≤∂u/∂ν< ∂(2u)/∂ν on∂Ω, (4.6)

Φλ,2u=Φλ in aC01(Ω)-neighborhood ofu0, sou0 is a local minimizer ofΦλ|C1 0(Ω), and hence also ofΦλby Brezis and Nirenberg [3] for p=2 and by Guo and Zhang [9] for

p >2. The mountain pass lemma now gives a second critical point as (1.4) implies that

Φλsatisfies the (PS) condition andΦλ(tu)→ −∞ast→ ∞.

Proof ofTheorem 1.5is similar and therefore omitted.

Acknowledgments

We would like to thank Professor Marco Degiovanni for showing us the proof of Proposition 2.1and Professor Mabel Cuesta and Professor Jean-Pierre Gossez for their helpful comments about the p-Laplacian. The first author was supported in part by the National Science Foundation. The second author was supported in part by the Na-tional Natural Science Foundation of China, Ky and Yu-Fen Fan Endowment of the AMS, Florida Institute of Technology, and the Humboldt Foundation.

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Kanishka Perera: Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA

E-mail address:[email protected]

Zhitao Zhang: Academy of Mathematics and Systems Science, Institute of Mathematics, Chinese Academy of Sciences, Beijing 100080, China